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NoetherNormalization :: noetherNormalization

noetherNormalization -- data for Noether normalization

Synopsis

Description

The computations performed in the routine noetherNormalization use a random linear change of coordinates, hence one should expect the output to change each time the routine is executed.
i1 : R = QQ[x_1..x_4];
i2 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);

o2 : Ideal of R
i3 : (f,J,X) = noetherNormalization I

               3                        2                      11 2         
o3 = (map(R,R,{-x  + x  + x , x , 4x  + -x  + x , x }), ideal (--x  + x x  +
               8 1    2    4   1    1   3 2    3   2            8 1    1 2  
     ------------------------------------------------------------------------
               3 3     17 2 2   2   3   3 2          2       2       2   2
     x x  + 1, -x x  + --x x  + -x x  + -x x x  + x x x  + 4x x x  + -x x x 
      1 4      2 1 2    4 1 2   3 1 2   8 1 2 3    1 2 3     1 2 4   3 1 2 4
     ------------------------------------------------------------------------
     + x x x x  + 1), {x , x })
        1 2 3 4         4   3

o3 : Sequence
The next example shows how when we use the lexicographical ordering, we can see the integrality of R/ f I over the polynomial ring in dim(R/I) variables:
i4 : R = QQ[x_1..x_5, MonomialOrder => Lex];
i5 : I = ideal(x_2*x_1-x_5^3, x_5*x_1^3);

o5 : Ideal of R
i6 : (f,J,X) = noetherNormalization I

               2                   10              4     1              
o6 = (map(R,R,{-x  + 2x  + x , x , --x  + x  + x , -x  + -x  + x , x }),
               5 1     2    5   1   9 1    2    4  5 1   4 2    3   2   
     ------------------------------------------------------------------------
            2 2                   3   8  3     24 2 2   12 2       24   3  
     ideal (-x  + 2x x  + x x  - x , ---x x  + --x x  + --x x x  + --x x  +
            5 1     1 2    1 5    2  125 1 2   25 1 2   25 1 2 5    5 1 2  
     ------------------------------------------------------------------------
     24   2     6     2     4      3       2 2      3
     --x x x  + -x x x  + 8x  + 12x x  + 6x x  + x x ), {x , x , x })
      5 1 2 5   5 1 2 5     2      2 5     2 5    2 5     5   4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10                                                          
     {-10} | 10x_1x_2x_5^6-96x_2^9x_5-320x_2^9+24x_2^8x_5^2+160x_2^8x_5-4x_2^
     {-9}  | 400x_1x_2^2x_5^3-30x_1x_2x_5^5+200x_1x_2x_5^4+288x_2^9-72x_2^8x_
     {-9}  | 160000x_1x_2^3+12000x_1x_2^2x_5^2+160000x_1x_2^2x_5+90x_1x_2x_5^
     {-3}  | 2x_1^2+10x_1x_2+5x_1x_5-5x_2^3                                  
     ------------------------------------------------------------------------
                                                                            
     7x_5^3-80x_2^7x_5^2+40x_2^6x_5^3-20x_2^5x_5^4+10x_2^4x_5^5+50x_2^2x_5^6
     5-160x_2^8+12x_2^7x_5^2+160x_2^7x_5-120x_2^6x_5^2+60x_2^5x_5^3-30x_2^4x
     5-300x_1x_2x_5^4+4000x_1x_2x_5^3+40000x_1x_2x_5^2-864x_2^9+216x_2^8x_5+
                                                                            
     ------------------------------------------------------------------------
                                                                           
     +25x_2x_5^7                                                           
     _5^4+200x_2^4x_5^3+2000x_2^3x_5^3-150x_2^2x_5^5+2000x_2^2x_5^4-75x_2x_
     720x_2^8-36x_2^7x_5^2-600x_2^7x_5+800x_2^7+360x_2^6x_5^2-1200x_2^6x_5-
                                                                           
     ------------------------------------------------------------------------
                                                                             
                                                                             
     5^6+500x_2x_5^5                                                         
     8000x_2^6-180x_2^5x_5^3+600x_2^5x_5^2+4000x_2^5x_5+80000x_2^5+90x_2^4x_5
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     ^4-300x_2^4x_5^3+4000x_2^4x_5^2+40000x_2^4x_5+800000x_2^4+60000x_2^3x_5^
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     2+1200000x_2^3x_5+450x_2^2x_5^5-1500x_2^2x_5^4+50000x_2^2x_5^3+600000x_2
                                                                             
     ------------------------------------------------------------------------
                                                                  |
                                                                  |
                                                                  |
     ^2x_5^2+225x_2x_5^6-750x_2x_5^5+10000x_2x_5^4+100000x_2x_5^3 |
                                                                  |

             5       1
o7 : Matrix R  <--- R
If noetherNormalization is unable to place the ideal into the desired position after a few tries, the following warning is given:
i8 : R = ZZ/2[a,b];
i9 : I = ideal(a^2*b+a*b^2+1);

o9 : Ideal of R
i10 : (f,J,X) = noetherNormalization I
--warning: no good linear transformation found by noetherNormalization

                                   2       2
o10 = (map(R,R,{a + b, a}), ideal(a b + a*b  + 1), {b})

o10 : Sequence
Here is an example with the option Verbose => true:
i11 : R = QQ[x_1..x_4];
i12 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);

o12 : Ideal of R
i13 : (f,J,X) = noetherNormalization(I,Verbose => true)
--trying random transformation: 1
--trying with basis element limit: 5
--trying with basis element limit: 20

                2                    9     9                      11 2  
o13 = (map(R,R,{-x  + 2x  + x , x , --x  + -x  + x , x }), ideal (--x  +
                9 1     2    4   1  10 1   8 2    3   2            9 1  
      -----------------------------------------------------------------------
                        1 3     41 2 2   9   3   2 2           2      9 2    
      2x x  + x x  + 1, -x x  + --x x  + -x x  + -x x x  + 2x x x  + --x x x 
        1 2    1 4      5 1 2   20 1 2   4 1 2   9 1 2 3     1 2 3   10 1 2 4
      -----------------------------------------------------------------------
        9   2
      + -x x x  + x x x x  + 1), {x , x })
        8 1 2 4    1 2 3 4         4   3

o13 : Sequence
The first number in the output above gives the number of linear transformations performed by the routine while attempting to place I into the desired position. The second number tells which BasisElementLimit was used when computing the (partial) Groebner basis. By default, noetherNormalization tries to use a partial Groebner basis. It does this by sequentially computing a Groebner basis with the option BasisElementLimit set to predetermined values. The default values come from the following list:{5,20,40,60,80,infinity}. To set the values manually, use the option LimitList:
i14 : R = QQ[x_1..x_4]; 
i15 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);

o15 : Ideal of R
i16 : (f,J,X) = noetherNormalization(I,Verbose => true,LimitList => {5,10})
--trying random transformation: 1
--trying with basis element limit: 5
--trying with basis element limit: 10

                8     6                                         11 2   6    
o16 = (map(R,R,{-x  + -x  + x , x , x  + 4x  + x , x }), ideal (--x  + -x x 
                3 1   7 2    4   1   1     2    3   2            3 1   7 1 2
      -----------------------------------------------------------------------
                  8 3     242 2 2   24   3   8 2       6   2      2      
      + x x  + 1, -x x  + ---x x  + --x x  + -x x x  + -x x x  + x x x  +
         1 4      3 1 2    21 1 2    7 1 2   3 1 2 3   7 1 2 3    1 2 4  
      -----------------------------------------------------------------------
          2
      4x x x  + x x x x  + 1), {x , x })
        1 2 4    1 2 3 4         4   3

o16 : Sequence
To limit the randomness of the coefficients, use the option RandomRange. Here is an example where the coefficients of the linear transformation are random integers from -2 to 2:
i17 : R = QQ[x_1..x_4];
i18 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);

o18 : Ideal of R
i19 : (f,J,X) = noetherNormalization(I,Verbose => true,RandomRange => 2)
--trying random transformation: 1
--trying with basis element limit: 5
--trying with basis element limit: 20

                                                                     2       
o19 = (map(R,R,{- 2x  - x  + x , x , 2x  - 2x  + x , x }), ideal (- x  - x x 
                    1    2    4   1    1     2    3   2              1    1 2
      -----------------------------------------------------------------------
                      3       2 2       3     2          2       2      
      + x x  + 1, - 4x x  + 2x x  + 2x x  - 2x x x  - x x x  + 2x x x  -
         1 4          1 2     1 2     1 2     1 2 3    1 2 3     1 2 4  
      -----------------------------------------------------------------------
          2
      2x x x  + x x x x  + 1), {x , x })
        1 2 4    1 2 3 4         4   3

o19 : Sequence

This symbol is provided by the package NoetherNormalization.

Ways to use noetherNormalization :